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=== Density of states ===
The 3D [[density of states]] (number of energy states, per energy per volume) of a non-interacting electron gas is given by:<ref group="Ashcroft & Mermin">{{
:<math>g(E) = \frac{m_e}{\pi^2\hbar^3}\sqrt{2m_eE} = \frac{3}{2}\frac{n}{E_{\rm F}}\sqrt{\frac{E}{E_{\rm F}}},</math>
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=== Heat capacity ===
{{Further|Electronic specific heat}}
One open problem in solid-state physics before the arrival of quantum mechanics was to understand the [[heat capacity]] of metals. While most solids had a constant [[volumetric heat capacity]] given by [[Dulong–Petit law]] of about <math>3nk_{\rm B}</math> at large temperatures, it did correctly predict its behavior at low temperatures. In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity.
The classical calculation using Drude's model, based on an ideal gas, provides a volumetric heat capacity given by
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:<math>c_V=\left(\frac{\partial u}{\partial T}\right)_{n}=\frac{\pi^2}{2}\frac{T}{T_{\rm F}} nk_{\rm B}</math>,
where the prefactor to <math>nk_B</math> is considerably smaller than the 3/2 found in <math display="inline">c^{\text{Drude}}_V</math>, about 100 times smaller at room temperature and much smaller at lower <math display="inline">T</math>.
Evidently, the electronic contribution alone does not predict the [[Dulong–Petit law]], i.e. the observation that the heat capacity of a metal is still constant at high temperatures. The free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice. Two famous quantum corrections include the [[Einstein solid]] model and the more refined [[Debye model]]. With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=49}}</ref>
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\end{matrix}\right.</math>
The free electron model is closer to the measured value of <math>L=2.44\times10^{-
However, Drude's mode predicts the wrong order of magnitude for the [[Seebeck coefficient]] (thermopower), which relates the generation of a potential difference by applying a temperature gradient across a sample <math>\nabla V =-S \nabla T</math>. This coefficient can be showed to be <math>S=-{c_{\rm V}}/{|ne|}</math>, which is just proportional to the heat capacity, so the Drude model predicts a constant that is hundred times larger than the value of the free electron model.<ref name=":7" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=23|ps=}}</ref> While the latter get as coefficient that is linear in temperature and provides much more accurate absolute values in the order of a few tens of
==Inaccuracies and extensions==
The free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:<ref name=":4" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=58-59}}</ref>
; Temperature dependence: The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. The thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fails to predict high temperature behaviour coming from ion motion and [[phonon]] scattering.
; Hall effect and magnetoresistance: The Hall coefficient has a constant value
; Directional: The conductivity of some metals can depend of the orientation of the sample with respect to the electric field. Sometimes even the electrical current is not parallel to the field. This possibility is not described because the model does not integrate the crystallinity of metals, i.e. the existence of a periodic lattice of ions.
; Diversity in the conductivity: Not all materials are [[electrical conductor]]s, some do not conduct electricity very well ([[Insulator (electricity)|insulators]]), some can conduct when impurities are added like [[semiconductor]]s. [[Semimetal]]s, with narrow conduction bands also exist. This diversity is not predicted by the model and can only by explained by analysing the [[valence and conduction bands]]. Additionally, electrons are not the only charge carriers in a metal, electron vacancies or [[Electron hole|holes]] can be seen as [[quasiparticle]]s carrying positive electric charge. Conduction of holes leads to an opposite sign for the Hall and Seebeck coefficients predicted by the model.
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;General
*{{cite book | last = Kittel | first = Charles | author-link=Charles Kittel | title = [[Introduction to Solid State Physics]] | ___location = University of Michigan | year = 1972|publisher=Wiley & Sons|isbn=978-0-471-49024-1}}
*{{cite book |
*{{cite book |
* {{cite book|last1=Ziman|first1=J.M.|title=Principles of the theory of solids|edition=2nd|publisher=Cambridge university press|year=1972|isbn=0-521-29733-8}}
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